How do I cut up a 3D object into separate pieces
Cell Fracture AddOn: Cell Fracture AddOn.  I'm pretty sure this is in Blender by default, but you may need to enable it in the User Preferences, then select the object, then bring up the Toolbar on the left of the 3D Window that you have the object selected in by pressing T, select Tools/Edit pane, then select Cell Fracture.

Or Fracture Modifier build (not an AddOn or plug-in) from Dennis Fassbaenders web site: Fracture Modifier Blender Build.  Explore that site too see if it's what you need.

Or do it all manually for complete control.